At home, I am praying with Sree Lakshmi Ganapathi god in idol form instead of praying Sree Ganapathi and Sree Lakshmi goddess in idol form. Is it correct? or is it recommended to keep the idol of Sree Lakshmi goddess and Sree Ganapathi idol and pray along with Sree Lakshmi Ganapathi. What is the recommendation?
I was confused whether should I comment or answer your question, but it's better I write this in the form of an answer.
Lakshmi-Ganesh is one of the 32 forms of Ganapati. And the rule is you cannot establish two idols of the same God, if you do you need to perform every puja twice, but its better not to keep same gods idol at one place. Keeping this in mind you can freely establish Lakshmi and Lakshmi-Ganesh, since in Lakshmi-Ganesha: Lakshmi is acting like a shakti and is part of Ganapati itself. While you must not keep Ganapati and Lakshmi-Ganapati together, the reason is they are both Ganapati's, although of diffrent forms they represent the same Devata.
If you wish you can keep photo frame of Ganapati along with Lakshimi-Ganapati. In my house itself I have Ganapati idol, and photo frames of Lakshimi-Ganapati and Lakshmi at one place.
Although certain texts say that keeping three causes problems:
गृहे लिक़द्य॑ ना्च्य॑ गणेशत्रितयं तथा । शद्धुद्रयं॑ तथा सूर्यों नाच्यों शक्ततत्रय तथा ॥ द्वे चक्रे द्वास्कायास्तु शालग्रामशिलाइयम तेषां तु पूजनेनैत उद्देग॑ प्राप्लुबाद गुही ॥
Performing the puja of two shiv linga, three Ganesha, two shakha, two sun, three durga murti, two gomati chakra, and two Shaligrama is inauspicious (brings Ashanti). From: here
But I would still restrict Ganapati to one per house.
- Never keep two Ganesha idols, its restricted to one per house.
- If you wish you can keep photo frames.
- You can keep Lakshmi separately in idol form.